From 4f2a703c1228e43284df359753201dddc6e3f5d9 Mon Sep 17 00:00:00 2001 From: didericis Date: Sat, 9 May 2026 15:45:51 -0400 Subject: [PATCH] Refute min-degree-5 plane diamond coloring conjecture at order 28 Adds search_min_degree_counterexample_comprehensive iterating Sage's planar_graphs generator with minimum_degree=5. Exhaustive enumeration through order 27 (456,967 maximal planar graphs of minimum degree at least 5) finds no counterexample to Conjecture 2.4. At order 28, three counterexamples are exhibited and verified via Sage's chromatic_number on the auxiliary graph, refuting the conjecture. Updates paper with the refutation theorem, the per-order census, a figure of one counterexample, and graph6 strings of the other two. Co-Authored-By: Claude Opus 4.7 --- .../min_degree_5_counterexample.png | Bin 0 -> 56300 bytes papers/plane_diamond_coloring/paper.aux | 9 ++- .../plane_diamond_coloring/paper.fdb_latexmk | 8 +-- papers/plane_diamond_coloring/paper.fls | 11 +++- papers/plane_diamond_coloring/paper.log | 57 ++++++++++-------- papers/plane_diamond_coloring/paper.pdf | Bin 223251 -> 268316 bytes papers/plane_diamond_coloring/paper.tex | 35 ++++++++++- 7 files changed, 82 insertions(+), 38 deletions(-) create mode 100755 papers/plane_diamond_coloring/min_degree_5_counterexample.png diff --git a/papers/plane_diamond_coloring/min_degree_5_counterexample.png b/papers/plane_diamond_coloring/min_degree_5_counterexample.png new file mode 100755 index 0000000000000000000000000000000000000000..df1996e3a431e6fd8df076ba47726340d50f2de4 GIT binary patch literal 56300 zcmaI8cRbeZ|2~e$o`ooTW)-0#GdtOrJ<8s)M(I$!7c9LI4ykMj;yQIaLZqsBu+LnD-zlTt%NLm!2I#&E8}PZ+qb 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+\newlabel{thm:mindeg5counterexample}{{3.5}{3}} +\bibcite{appel1977every}{1} +\bibcite{robertson1997four}{2} \bibcite{mckaygraph6}{3} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces One of three known smallest counterexamples to Conjecture\nonbreakingspace 3.4\hbox {}: a maximal planar graph on $28$ vertices with minimum degree $5$ admitting no plane diamond coloring.}}{4}{}\protected@file@percent } +\newlabel{fig:mindeg5counterexample}{{2}{4}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent } \gdef \@abspage@last{4} diff --git a/papers/plane_diamond_coloring/paper.fdb_latexmk b/papers/plane_diamond_coloring/paper.fdb_latexmk index 71fadc2..8455449 100644 --- a/papers/plane_diamond_coloring/paper.fdb_latexmk +++ b/papers/plane_diamond_coloring/paper.fdb_latexmk @@ -1,6 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778347081 "/Users/didericis/Code/math-research/papers/plane_diamond_coloring/paper.tex" "paper.pdf" "paper" 1778347082 - "/Users/didericis/Code/math-research/papers/plane_diamond_coloring/paper.tex" 1778347081 7877 4504fbad3843f8cebeb3943b86c7fcee "" +["pdflatex"] 1778355822 "paper.tex" "paper.pdf" "paper" 1778355822 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -62,8 +61,9 @@ "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" "counterexample.png" 1778345471 31680 49acd944c061e194d20aac1d5cd8ef86 "" - "paper.aux" 1778347082 1030 fa34fda9f562babd4bf0c5868b521a80 "pdflatex" - "paper.tex" 1778347081 7877 4504fbad3843f8cebeb3943b86c7fcee "" + "min_degree_5_counterexample.png" 1778355623 56300 15b8c2d55f1dc6b52f306af893b9fd60 "" + "paper.aux" 1778355822 1755 52c9cc0d4e80337d3db689115d06b369 "pdflatex" + "paper.tex" 1778355812 12440 9687ca443e7a03568da7d185c37d3adf "" (generated) "paper.aux" "paper.log" diff --git a/papers/plane_diamond_coloring/paper.fls b/papers/plane_diamond_coloring/paper.fls index aca8a20..0fe1f76 100644 --- a/papers/plane_diamond_coloring/paper.fls +++ b/papers/plane_diamond_coloring/paper.fls @@ -2,7 +2,7 @@ PWD /Users/didericis/Code/math-research/papers/plane_diamond_coloring INPUT /usr/local/texlive/2022/texmf.cnf INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt -INPUT /Users/didericis/Code/math-research/papers/plane_diamond_coloring/paper.tex +INPUT paper.tex OUTPUT paper.log INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsart.cls @@ -224,16 +224,21 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm +OUTPUT paper.pdf +INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm -OUTPUT paper.pdf -INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map INPUT ./counterexample.png INPUT ./counterexample.png INPUT counterexample.png INPUT ./counterexample.png INPUT ./counterexample.png +INPUT ./min_degree_5_counterexample.png +INPUT ./min_degree_5_counterexample.png +INPUT min_degree_5_counterexample.png +INPUT ./min_degree_5_counterexample.png +INPUT ./min_degree_5_counterexample.png INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm INPUT paper.aux INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb diff --git a/papers/plane_diamond_coloring/paper.log b/papers/plane_diamond_coloring/paper.log index fa391b6..984ab59 100644 --- a/papers/plane_diamond_coloring/paper.log +++ b/papers/plane_diamond_coloring/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 13:44 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 15:43 entering extended 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planar g Every maximal planar graph $G$ of minimum degree at least $5$ has a plane diamond coloring. \end{conjecture} -\begin{remark} -We have verified Conjecture~\ref{conj:mindeg5} computationally for all maximal planar graphs of minimum degree at least $5$ and order at most $N$, by exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator and the auxiliary-graph reduction described in the proof of Theorem~\ref{thm:counterexample}. No counterexample has been found. -\end{remark} +\begin{theorem} \label{thm:mindeg5counterexample} +Conjecture~\ref{conj:mindeg5} is false. The smallest counterexamples have order $28$, and every maximal planar graph of minimum degree at least $5$ and order at most $27$ admits a plane diamond coloring. +\end{theorem} + +\begin{proof} +By exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator (with \texttt{minimum\_connectivity=3}, \texttt{maximum\_face\_size=3}, and \texttt{minimum\_degree=5}) and the auxiliary-graph reduction described in the proof of Theorem~\ref{thm:counterexample}, every maximal planar graph of minimum degree at least $5$ and order in $\{12, 13, \dots, 27\}$ admits a plane diamond coloring. The numbers of such triangulations at orders $12, 13, \dots, 27$ are +\[ + 1,\ 0,\ 1,\ 1,\ 3,\ 4,\ 12,\ 23,\ 73,\ 192,\ 651,\ 2070,\ 7290,\ 25381,\ 91441,\ 329824, +\] +totalling $456{,}967$ graphs, none of which is a counterexample. + +At order $28$, however, counterexamples do exist. The graph in Figure~\ref{fig:mindeg5counterexample} is one such, with canonical graph6 string +\[ + \verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??B???|C?CIG?GIA?iD@?TPC?VQG_Bi+. +\] +It has $|V| = 28$, $|E| = 78 = 3 \cdot 28 - 6$, minimum degree $5$, and chromatic number $4$. Two further counterexamples at order $28$ have canonical graph6 strings +\[ + \verb+[?`???I@PCAG????@COGaGA_OD?DD?Aa_AII?PPV???Y??@ii?ATT?@T?T@agAgX+ +\] +and +\[ + \verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??BIA??gG?PC?IPC?Ig_?tIG?TO??F~+ . +\] +Direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies $\chi(H_u) > 4$ for every $u$ in each of these graphs. +\end{proof} + +\begin{figure}[htbp] + \centering + \includegraphics[width=0.45\textwidth]{min_degree_5_counterexample.png} + \caption{One of three known smallest counterexamples to Conjecture~\ref{conj:mindeg5}: a maximal planar graph on $28$ vertices with minimum degree $5$ admitting no plane diamond coloring.} + \label{fig:mindeg5counterexample} +\end{figure} \begin{thebibliography}{9}